Editing Heterodyne

{{Short description|Signal processing technique}}
{{About|waveform manipulation}}
{{Use mdy dates|date=April 2018}}
[[Image:IdealMixer.svg|thumb|upright=1.5|Frequency mixer symbol used in schematic diagrams]]

A '''heterodyne''' is a [[signal]] [[frequency]] that is created by combining or mixing two other frequencies using a [[signal processing]] technique called ''heterodyning'', which was invented by Canadian inventor-engineer [[Reginald Fessenden]].<ref name="Cooper2001">{{cite book|author=Christopher E. Cooper|title=Physics|url=https://books.google.com/books?id=lEHsmDZigYMC&pg=PA25|date=January 2001|publisher=Fitzroy Dearborn Publishers|isbn=978-1-57958-358-3|pages=25–}}</ref><ref name="Naval">{{cite book | last = United States Bureau of Naval Personnel | author-link = Bureau of Naval Personnel | title = Basic Electronics | publisher = Courier Dover | date = 1973 | location = USA | page = 338 | url = https://books.google.com/books?id=o6pn1Pdas1UC&q=heterodyne+mixer+modulation&pg=PA338 | isbn = 978-0-486-21076-6}}</ref><ref name="Graf">{{cite book | last = Graf | first = Rudolf F. | title = Modern dictionary of electronics | edition=7th | publisher = Newnes | date = 1999 | location = USA | page = 344 | url = https://books.google.com/books?id=uah1PkxWeKYC&q=heterodyning+frequencies&pg=PA344 | isbn = 978-0-7506-9866-5}}</ref> Heterodyning is used to shift signals from one [[frequency range]] into another, and is also involved in the processes of [[modulation]] and [[demodulation]].<ref name="Naval" /><ref name="Horowitz">{{cite book | last1 = Horowitz | first1 = Paul | author1-link = Paul Horowitz |last2 = Hill | first2 = Winfield | author2-link = Winfield Hill | title = The Art of Electronics | edition = 2nd | publisher = Cambridge University Press | date = 1989 | location = London | pages = 885, 897 | url = https://books.google.com/books?id=bkOMDgwFA28C&q=heterodyning | isbn = 978-0-521-37095-0}}</ref> The two input frequencies are combined in a [[linear circuit|nonlinear]] signal-processing device such as a [[vacuum tube]], [[transistor]], or [[diode]], usually called a ''[[frequency mixer|mixer]]''.<ref name="Naval" />

In the most common application, two signals at frequencies {{math|''f''<sub>1</sub>}} and  {{math|''f''<sub>2</sub>}} are mixed, creating two new signals, one at the sum of the two frequencies {{math|''f''<sub>1</sub>&nbsp;+&nbsp;''f''<sub>2</sub>}}, and the other at the difference between the two frequencies {{math|''f''<sub>1</sub>&nbsp;−&nbsp;''f''<sub>2</sub>}}.<ref name="Graf" /> The new signal frequencies are called ''heterodynes''. Typically, only one of the heterodynes is required and the other signal is [[filter (signal processing)|filtered]] out of the output of the mixer. Heterodyne frequencies are related to the phenomenon of "[[beat (acoustics)|beat]]s" in acoustics.<ref name="Naval" /><ref name="Strange">{{cite book | last2 = Strange | first2 = Patricia | last1 = Strange | first1 = Allen | author1-link = Allen Strange | title = The Contemporary Violin: Extended Performance Techniques | publisher = Scarecrow Press | date = 2003 | page = 216 | url = https://books.google.com/books?id=HZtGs5_z4zIC&q=heterodyning+beat&pg=PA216 | isbn = 978-0-520-22409-4}}</ref><ref name="Ingard">{{cite book | last1 = Ingard | first1 = Uno | title = Acoustics | publisher = Jones and Bartlett | date = 2008 | pages = 18–21 | url = https://books.google.com/books?id=LD_wlUs2Fz0C&q=heterodyning+beat&pg=PA20 | isbn = 978-1-934015-08-7 }}</ref>

A major application of the heterodyne process is in the [[superheterodyne receiver|superheterodyne radio receiver]] circuit, which is used in virtually all modern radio receivers.

==History==
[[File:Heterodyne radio receiver circuit 1920.png|thumb|Fessenden's heterodyne radio receiver circuit. The incoming radio frequency and local oscillator frequency mix in the crystal diode detector.]]

In 1901, [[Reginald Fessenden]] demonstrated a [[direct-conversion receiver]] or [[beat receiver]] as a method of making [[continuous wave]] [[radiotelegraphy]] signals audible.<ref>[http://www.nonstopsystems.com/radio/pdf-hell/article-IRE-7-1959.pdf Discussion of A History of Some Foundations of Modern Radio-Electronic Technology], Comments by Lloyd Espenschied, Proceedings of the IRE, July, 1959 (Vol. 47, No. 7), pp. 1254, 1256. ''Critique.'' ". . . the roots of our modern technology trace back generally to sources other than the Hammond Laboratory."  ''Comment.'' Many of the roots that nourished the work of the Hammond group and its contemporaries were recorded in our paper: the pioneering work of Wilson and Evans, Tesla, Shoemaker, in basic radiodynamics; . . . of Tesla and Fessenden leading to the development of basic intermediate frequency circuitry.</ref> Fessenden's receiver did not see much application because of its local oscillator's stability problem.   A stable yet inexpensive local oscillator was not available until [[Lee de Forest]] invented the [[triode|triode vacuum tube]] oscillator.<ref name=nahin1>{{Harvnb|Nahin|2001|p=91}}, stating "Fessenden's circuit was ahead of its time, however, as there simply was no technology available then with which to build the required local oscillator with the necessary frequency stability." Figure 7.10 shows a simplified 1907 heterodyne detector.</ref> In a 1905 patent, Fessenden stated that the frequency stability of his local oscillator was one part per thousand.<ref>{{Harvnb|Fessenden|1905|p=4}}</ref>

In radio telegraphy, the characters of text messages are translated into the short duration dots and long duration dashes of [[Morse code]] that are broadcast as radio signals. [[Radio telegraphy]] was much like ordinary [[telegraphy]]. One of the problems was building high power transmitters with the technology of the day. Early transmitters were [[spark gap transmitter]]s. A mechanical device would make sparks at a fixed but audible rate; the sparks would put energy into a resonant circuit that would then ring at the desired transmission frequency (which might be 100&nbsp;kHz). This ringing would quickly decay, so the output of the transmitter would be a succession of [[damped wave]]s. When these damped waves were received by a simple detector, the operator would hear an audible buzzing sound that could be transcribed back into alpha-numeric characters.<!-- spark gap transmitters had a lot of splatter. Californian's operator was asked to shut down early so Titanic could work Cape Race. Consequently, California's radio room was not manned when Titanic sent out its distress signal. http://www.environmentalhistory.org/revcomm/features/radio-and-the-titanic/-->

With the development of the [[arc converter]] radio transmitter in 1904, [[continuous wave]] (CW) modulation began to be used for radiotelegraphy.  CW Morse code signals are not amplitude modulated, but rather consist of bursts of sinusoidal carrier frequency.  When CW signals are received by an AM receiver, the operator does not hear a sound. The direct-conversion (heterodyne) detector was invented to make continuous wave radio-frequency signals audible.<ref>{{cite book |last1=Ashley |first1=Charles Grinnell |last2=Heyward |first2=Charles Brian |title=Wireless Telegraphy and Wireless Telephony |publisher=American School of Correspondence |year=1912 |location=Chicago |pages=103/15–104/16 |url=https://archive.org/details/wirelesstelegra00haywgoog }}</ref>

The "heterodyne" or "beat" receiver has a [[local oscillator]] that produces a radio signal adjusted to be close in frequency to the incoming signal being received.  When the two signals are mixed, a "beat" frequency equal to the difference between the two frequencies is created.  Adjusting the local oscillator frequency correctly puts the beat frequency in the [[audio signal|audio]] range, where it can be heard as a tone in the receiver's [[earphone]]s whenever the transmitter signal is present. Thus the Morse code "dots" and "dashes" are audible as beeping sounds. This technique is still used in radio telegraphy, the local oscillator now being called the [[beat frequency oscillator]] or BFO. Fessenden coined the word ''heterodyne'' from the Greek roots ''hetero-'' "different", and ''dyn-'' "power" (cf. [[Wiktionary:δύναμις|δύναμις or dunamis]]).<ref>[https://books.google.com/books?id=49jgQvbrvCUC&dq=heterodyne+Fessenden+coined&pg=PA370 Tapan K. Sarkar, History of wireless, page 372]</ref>

===Superheterodyne receiver===
[[Image:Superheterodyne receiver block diagram 2.svg|thumb|upright=1.6|Block diagram of a typical superheterodyne receiver.  <span style="color:red;">Red</span> parts are those that handle the incoming radio frequency (RF) signal; <span style="color:green;">green</span> are parts that operate at the intermediate frequency (IF), while <span style="color:blue;">blue</span> parts operate at the modulation (audio) frequency.]]

An important and widely used application of the heterodyne technique is in the [[superheterodyne receiver]] (superhet). In the typical superhet, the incoming [[radio frequency]] signal from the antenna is mixed (heterodyned) with a signal from a local oscillator (LO) to produce a lower fixed frequency signal called the [[intermediate frequency]] (IF) signal. The IF signal is amplified and filtered and then applied to a [[detector (radio)|detector]] that extracts the audio signal; the audio is ultimately sent to the receiver's loudspeaker.

The superheterodyne receiver has several advantages over previous receiver designs. One advantage is easier tuning; only the RF filter and the LO are tuned by the operator; the fixed-frequency IF is tuned ("aligned") at the factory and is not adjusted. In older designs such as the [[tuned radio frequency receiver]] (TRF), all of the receiver stages had to be simultaneously tuned. In addition, since the IF filters are fixed-tuned, the receiver's selectivity is the same across the receiver's entire frequency band. Another advantage is that the IF signal can be at a much lower frequency than the incoming radio signal, and that allows each stage of the IF amplifier to provide more gain. To first order, an amplifying device has a fixed [[gain-bandwidth product]]. If the device has a gain-bandwidth product of 60&nbsp;MHz, then it can provide a voltage gain of 3 at an RF of 20&nbsp;MHz or a voltage gain of 30 at an IF of 2&nbsp;MHz. At a lower IF, it would take fewer gain devices to achieve the same gain. The [[regenerative radio receiver]] obtained more gain out of one gain device by using positive feedback, but it required careful adjustment by the operator; that adjustment also changed the selectivity of the regenerative receiver. The superheterodyne provides a large, stable gain and constant selectivity without troublesome adjustment.

The superior superheterodyne system replaced the earlier TRF and regenerative receiver designs, and since the 1930s most commercial radio receivers have been superheterodynes.

==Applications==
Heterodyning, also called ''frequency conversion'', is used very widely in [[communications engineering]] to generate new frequencies and move information from one frequency channel to another. Besides its use in the superheterodyne circuit found in almost all radio and television receivers, it is used in [[radio transmitter]]s, [[modem]]s, [[satellite]] communications and set-top boxes, [[radar]], [[radio telescope]]s, [[telemetry]] systems, cell phones, cable television converter boxes and [[Cable television headend|headends]], [[microwave relay]]s, [[metal detector]]s, [[atomic clock]]s, and military [[electronic countermeasure]] (jamming) systems.

===Up and down converters===
In large scale [[telecommunication network]]s such as [[telephone network]] trunks, [[microwave relay]] networks, cable television systems, and [[communication satellite]] links, large [[bandwidth (signal processing)|bandwidth]] capacity links are shared by many individual communication channels by using heterodyning to move the frequency of the individual signals up to different frequencies, which share the channel. This is called [[frequency division multiplexing]] (FDM).

For example, a [[coaxial cable]] used by a cable television system can carry 500 television channels at the same time because each one is given a different frequency, so they do not interfere with one another. At the cable source or [[Cable television headend|headend]], electronic upconverters convert each incoming television channel to a new, higher frequency.  They do this by mixing the television signal frequency, ''f<sub>CH</sub>'' with a [[local oscillator]] at a much higher frequency {{math|''f<sub>LO</sub>''}}, creating a heterodyne at the sum {{math|''f<sub>CH</sub>''&nbsp;+&nbsp;''f<sub>LO</sub>''}}, which is added to the cable. At the consumer's home, the cable [[set top box]] has a downconverter that mixes the incoming signal at frequency {{math|''f<sub>CH</sub>''&nbsp;+&nbsp;''f<sub>LO</sub>''}} with the same local oscillator frequency {{math|''f<sub>LO</sub>''}} creating the difference heterodyne frequency, converting the television channel back to its original frequency: {{math|1=(''f<sub>CH</sub>''&nbsp;+&nbsp;''f<sub>LO</sub>'')&nbsp;−&nbsp;''f<sub>LO</sub>'' =&nbsp;''f<sub>CH</sub>''}}.  Each channel is moved to a different higher frequency.  The original lower basic frequency of the signal is called the [[baseband]], while the higher channel it is moved to is called the [[passband]].

===Analog videotape recording===
<!-- [[Color under]] redirects here. Please fix it if the name of this section changes :) -->
Many analog [[videotape]] systems rely on a downconverted color subcarrier to record color information in their limited bandwidth. These systems are referred to as "heterodyne systems" or "color-under systems". For instance, for [[NTSC]] video systems, the [[VHS]] (and [[S-VHS]]) recording system converts the color subcarrier from the NTSC standard 3.58&nbsp;MHz to ~629&nbsp;kHz.<ref name = lionlamb>[http://www.lionlmb.org/quad/format.html#12incomposite Videotape formats using {{convert|1/2|in|mm|adj=mid|-wide}} tape] {{Webarchive|url=https://web.archive.org/web/20060616104342/http://lionlmb.org/quad/format.html#12incomposite |date=June 16, 2006 }} ; Retrieved 2007-01-01</ref> [[PAL]] VHS color subcarrier is similarly downconverted (but from 4.43&nbsp;MHz). The now-obsolete 3/4" [[U-matic]] systems use a heterodyned ~688&nbsp;kHz subcarrier for NTSC recordings (as does [[Sony]]'s [[Betamax]], which is at its basis a 1/2″ consumer version of U-matic), while PAL U-matic decks came in two mutually incompatible varieties, with different subcarrier frequencies, known as Hi-Band and Low-Band. Other videotape formats with heterodyne color systems include [[Video-8]] and [[Hi8]].<ref name = poynton>{{cite book|first=Poynton |last=Charles |author-link=Charles Poynton |title=Digital Video and HDTV: Algorithms and Interfaces |year=2003 |publisher=Morgan Kaufmann Publishers |location=San Francisco |pages=582–3 |isbn=978-1-55860-792-7 |url=https://books.google.com/books?id=ra1lcAwgvq4C}}</ref>

The heterodyne system in these cases is used to convert quadrature phase-encoded and amplitude modulated sine waves from the broadcast frequencies to frequencies recordable in less than 1&nbsp;MHz bandwidth. On playback, the recorded color information is heterodyned back to the standard subcarrier frequencies for display on televisions and for interchange with other standard video equipment.

Some U-matic (3/4″) decks feature 7-pin mini-[[DIN connector]]s to allow dubbing of tapes without conversion, as do some industrial VHS, S-VHS, and Hi8 recorders.

===Music synthesis===
The [[theremin]], an [[electronic musical instrument]], traditionally uses the heterodyne principle to produce a variable [[audio frequency]] in response to the movement of the musician's hands in the vicinity of one or more antennae, which act as capacitor plates. The output of a fixed radio frequency oscillator is mixed with that of an oscillator whose frequency is affected by the [[variable capacitance]] between the antenna and the musician's hand as it is moved near the pitch control antenna. The difference between the two oscillator frequencies produces a tone in the audio range.

The [[ring modulator]] is a type of [[frequency mixer]] incorporated into some synthesizers or used as a stand-alone audio effect.

===Optical heterodyning===
[[Optical heterodyne detection]] (an area of active research) is an extension of the heterodyning technique to higher (visible) frequencies. Guerra<ref>{{Cite journal |last=Guerra |first=John M. |date=1995-06-26 |title=Super-resolution through illumination by diffraction-born evanescent waves |url=http://aip.scitation.org/doi/10.1063/1.113814 |journal=Applied Physics Letters |language=en |volume=66 |issue=26 |pages=3555–3557 |doi=10.1063/1.113814 |bibcode=1995ApPhL..66.3555G |issn=0003-6951}}</ref> (1995) first published the results of what he called a "form of optical heterodyning" in which light patterned by a 50 nm pitch grating illuminated a second grating of pitch 50 nm, with the gratings rotated with respect to each other by the angular amount needed to achieve magnification. Although the illuminating wavelength was 650 nm, the 50 nm grating was easily resolved. This showed a nearly 5-fold improvement over the Abbe resolution limit of 232 nm that should have been the smallest obtained for the numerical aperture and wavelength used. This super-resolution microscopic imaging through optical heterodyning later came to be know by many as "structured illumination microscopy".  

In addition to super-resolution optical microscopy, optical heterodyning could greatly improve [[optical modulator]]s, increasing the density of information carried by [[optical fiber]]s. It is also being applied in the creation of more accurate [[atomic clock]]s based on directly measuring the frequency of a laser beam.{{refn|group=notes|See [[NIST]] subtopic 9.07.9-4.R for a description of research on one system to do this.<ref>[http://tsapps.nist.gov/ts_sbir/sbirrss/index.cfm?action=contractdetails&id=78 Contract Details: Robust Nanopopous Ceramic Microsensor Platform<!-- Bot generated title -->]</ref><ref>[http://tsapps.nist.gov/ts_sbir/sbirrss/index.cfm?action=contractdetails&id=147 Contract Details: High Pulsed Power Varactor Multipliers for Imaging<!-- Bot generated title -->]</ref>}}

Since optical frequencies are far beyond the manipulation capacity of any feasible electronic circuit, all visible frequency photon detectors are inherently energy detectors not oscillating electric field detectors. However, since energy detection is inherently "[[square-law detector|square-law]]" detection, it intrinsically mixes any optical frequencies present on the detector. Thus, sensitive detection of specific optical frequencies necessitates optical heterodyne detection, in which two different (close by) wavelengths of light illuminate the detector so that the oscillating electrical output corresponds to the difference between their frequencies. This allows extremely narrow band detection (much narrower than any possible color filter can achieve) as well as precision measurements of phase and frequency of a light signal relative to a reference light source, as in a [[laser Doppler vibrometer]].

This phase sensitive detection has been applied for Doppler measurements of wind speed, and imaging through dense media. The high sensitivity against background light is especially useful for [[lidar]].

In [[optical Kerr effect]] (OKE) spectroscopy, optical heterodyning of the OKE signal and a small part of the probe signal produces a mixed signal consisting of probe, heterodyne OKE-probe and homodyne OKE signal. The probe and homodyne OKE signals can be filtered out, leaving the heterodyne frequency signal for detection.

Heterodyne detection is often used in [[interferometry]] but usually confined to single point detection rather than widefield interferometry, however, widefield heterodyne interferometry is possible using a special camera.<ref>{{cite journal|last=Patel|first=R.|author2=Achamfuo-Yeboah, S. |author3=Light R.|author4=Clark M.|title=Widefield heterodyne interferometry using a custom CMOS modulated light camera|journal=Optics Express|date=2011|volume=19|issue=24|pages=24546–24556|url=https://www.osapublishing.org/oe/abstract.cfm?uri=oe-19-24-24546|doi=10.1364/oe.19.024546|pmid=22109482|bibcode=2011OExpr..1924546P |doi-access=free}}</ref>   Using this technique which a reference signal extracted from a single pixel it is possible to build a highly stable widefield heterodyne interferometer by removing the piston phase component
caused by [[microphonics]] or vibrations of the optical components or object.<ref>{{cite journal|last=Patel|first=R.|author2=Achamfuo-Yeboah, S. |author3=Light R.|author4=Clark M.|title=Ultrastable heterodyne interferometer system using a CMOS modulated light camera|journal=Optics Express|date=2012|volume=20|issue=16|pages=17722–17733|url=https://www.osapublishing.org/oe/abstract.cfm?uri=oe-20-16-17722|doi=10.1364/oe.20.017722|pmid=23038324|bibcode=2012OExpr..2017722P |doi-access=free}}</ref>

==Mathematical principle==
Heterodyning is based on the [[trigonometric identity]]:

:<math>(\cos\theta_1) (\cos\theta_2) = \tfrac{1}{2}\cos(\theta_1 - \theta_2) + \tfrac{1}{2}\cos( \theta_1 + \theta_2)</math>

The product on the left hand side represents the multiplication ("mixing") of a [[sine wave]] with another sine wave (both produced by [[cosine]] functions). The right hand side shows that the resulting signal is the sum of two [[sinusoidal]] terms, one at the sum of the two original frequencies, and one at the difference, which can be dealt with separately, since their (large) frequency difference makes it easy to cleanly filter out one signal's frequency, while leaving the other signal unchanged.

Using this trigonometric identity, the result of multiplying two cosine wave signals <math>\ \cos\left(2 \pi f_1 t\right)\ </math> and <math>\ \cos\left(2 \pi f_2 t\right)\ </math> at different frequencies <math>\ f_1\ </math> and <math>\ f_2\ </math> can be calculated:

:<math>\cos(2 \pi f_1 t) \cos(2 \pi f_2 t) = \tfrac{1}{2}\cos[2 \pi ( f_1 - f_2) t ] + \tfrac{1}{2}\cos[ 2 \pi ( f_1 + f_2) t ]</math>

The result is the sum of two sinusoidal signals, one at the sum {{math|''f''<sub>1</sub>&nbsp;+&nbsp;''f''<sub>2</sub>}} and one at the difference {{math|''f''<sub>1</sub>&nbsp;−&nbsp;''f''<sub>2</sub>}} of the original frequencies.

===Mixer===
The two signals are combined in a device called a ''[[Frequency mixer|mixer]]''.  As seen in the previous section, an ideal mixer would be a device that multiplies the two signals.  Some widely used mixer circuits, such as the [[Gilbert cell]], operate in this way, but they are limited to lower frequencies.  However, any ''[[linear circuit|nonlinear]]'' electronic component also multiplies signals applied to it, producing heterodyne frequencies in its output—so a variety of nonlinear components serve as mixers. A nonlinear component is one in which the output current or voltage is a [[Linear function|nonlinear function]] of its input. Most circuit elements in communications circuits are designed to be [[Linear circuit|linear]]. This means they obey the [[superposition principle]]; if <math>\ F(v)\ </math> is the output of a linear element with an input of <math>\ v\ </math>:

:<math>\ F(v_1 + v_2) = F(v_1) + F(v_2)\ </math>

So if two sine wave signals at frequencies {{math|''f''<sub>1</sub>}} and {{math|''f''<sub>2</sub>}} are applied to a linear device, the output is simply the sum of the outputs when the two signals are applied separately with no product terms. Thus, the function <math>F</math> must be nonlinear to create mixer products.  A perfect multiplier only produces mixer products at the sum and difference frequencies {{math|(''f''<sub>1</sub>&nbsp;±&nbsp;''f''<sub>2</sub>)}}, but more general nonlinear functions produce higher order mixer products: {{math|''n''&sdot;''f''<sub>1</sub>&nbsp;+&nbsp;''m''&sdot;''f''<sub>2</sub>}} for integers {{math|''n''}} and {{math|''m''}}.  Some mixer designs, such as double-balanced mixers, suppress some high order undesired products, while other designs, such as [[harmonic mixer]]s exploit high order differences.

Examples of nonlinear components that are used as mixers are [[vacuum tube]]s and [[transistor]]s biased near cutoff ([[class C amplifier|class C]]), and [[diode]]s. [[Magnetic core|Ferromagnetic core]] [[inductor]]s driven into [[saturation (magnetic)|saturation]] can also be used at lower frequencies. In [[nonlinear optics]], crystals that have nonlinear characteristics are used to mix [[laser]] light beams to create [[Optical heterodyne detection|optical heterodyne frequencies]].

===Output of a mixer===
To demonstrate mathematically how a nonlinear component can multiply signals and generate heterodyne frequencies, the nonlinear function <math>F</math> can be expanded in a [[power series]] ([[MacLaurin series]]):

:<math>\ F(v) = \alpha_1 v + \alpha_2 v^2 + \alpha_3 v^3 + \cdots\ </math>

To simplify the math, the higher order terms above {{math|''α''<sub>2</sub>}} are indicated by an ellipsis (<math>\ \cdots\ </math>) and only the first terms are shown. Applying the two sine waves at frequencies {{math|1=''ω''<sub>1</sub> =&nbsp;2{{pi}}''f''<sub>1</sub>}} and {{math|1=''ω''<sub>2</sub> =&nbsp;2{{pi}}''f''<sub>2</sub>}} to this device:

:<math>\ v_\mathsf{out} = F\Bigl(A_1 \cos(\omega_1 t) + A_2 \cos(\omega_2 t) \Bigr)\ </math>

:<math>\ v_\mathsf{out} = \alpha_1 \Bigl( A_1 \cos(\omega_1 t) + A_2 \cos(\omega_2 t) \Bigr) + \alpha_2 \Bigl( A_1 \cos(\omega_1 t) + A_2 \cos(\omega_2 t) \Bigr)^2 + \cdots\ </math>

:<math>\ v_\mathsf{out} = \alpha_1 \Bigl( A_1 \cos(\omega_1 t) + A_2 \cos(\omega_2 t)\Bigr) + \alpha_2 \Bigl( A_1^2 \cos^2(\omega_1 t) + 2 A_1 A_2 \cos(\omega_1 t)\ \cos(\omega_2 t) + A_2^2 \cos^2(\omega_2 t) \Bigr) + \cdots\ </math>

It can be seen that the second term above contains a product of the two sine waves. Simplifying with [[trigonometric identity|trigonometric identities]]:

:<math>
\begin{align}
v_\mathsf{out} = {} & \alpha_1 \Bigl( A_1 \cos(\omega_1 t) + A_2 \cos(\omega_2 t) \Bigr) \\
& {} + \alpha_2 \Bigl( \tfrac{1}{2} A_1^2 [ 1 + \cos(2 \omega_1 t) ] + A_1 A_2 [\cos(\omega_1 t - \omega_2 t ) + \cos (\omega_1 t + \omega_2 t) ] + \tfrac{1}{2} A_2^2 [ 1 + \cos(2 \omega_2 t) ] \Bigr) + \cdots
\end{align}
</math>

Which leaves the two heterodyne frequencies among the many terms:
:<math>\ v_\mathsf{out} = \cdots + \alpha_2 A_1 A_2 \cos (\omega_1 - \omega_2 )t + \alpha_2 A_1 A_2 \cos (\omega_1 + \omega_2 ) t + \cdots\ </math>
along with many other terms not shown.

In addition to components with frequencies at the sum {{math|''ω''<sub>1</sub>&nbsp;+&nbsp;''ω''<sub>2</sub>}} and difference {{math|''ω''<sub>1</sub>&nbsp;−&nbsp;''ω''<sub>2</sub>}} of the two original frequencies, shown above, the output also contains sinusoidal terms at the original frequencies and terms at multiples of the original frequencies {{nobr|{{math|2 ''ω''<sub>1</sub>}} ,}} {{nobr|{{math|2 ''ω''<sub>2</sub>}} ,}} {{nobr|{{math|3 ''ω''<sub>1</sub>}} ,}} {{nobr|{{math|3 ''ω''<sub>2</sub>}} ,}} etc., called ''[[harmonics]]''. It also contains much more complicated terms at frequencies of {{nobr|{{math|''M ω''<sub>1</sub> + ''N ω''<sub>2</sub>}} ,}} called [[intermodulation product]]s. These unwanted frequencies, along with the unwanted heterodyne frequency, must be removed from the mixer output by an [[electronic filter]], to leave the desired heterodyne frequency.

==Notes==
{{reflist|group=notes}}

==See also==
* [[Electroencephalography]]
* [[Homodyne]]
* [[Intermodulation]] – a problem with strong higher-order terms produced in some non-linear mixers
* [[Transverter]]

== References ==
=== Citations ===
{{Reflist|30em}}

=== General and cited references ===
* {{Cite patent |inventor-last=Fessenden |inventor-first=Reginald A. |inventorlink=Reginald Fessenden |title=Electric Signaling Apparatus |country-code=US |patent-number=1050441 |issue-date=January 14, 1913 |publication-date=July 27, 1905 |ref=CITEREFFessenden1905 }}
* {{Cite book |last=Glinsky |first=Albert |author-link=Albert Glinsky |year=2000 |title=Theremin: Ether Music and Espionage |url=https://archive.org/details/thereminethermus00glin |url-access=registration |location=Urbana, IL |publisher=University of Illinois Press |isbn=978-0-252-02582-2}}
* {{Cite book |last=Nahin |first=Paul J. |author-link=Paul J. Nahin |year=2001 |title=The Science of Radio with Matlab and Electronics Workbench Demonstrations |url=https://books.google.com/books?id=V1GBW6UD4CcC |edition=Second |location=New York |publisher=Springer-Verlag, AIP Press |isbn=978-0-387-95150-8}}

== Further reading ==
* {{Cite magazine |first=John V. L. |last=Hogan |date=April 1921 |title=The Heterodyne Receiver |url=https://archive.org/details/electricjournal18elecuoft |journal=Electric Journal |volume=18 |page=116}}
* {{Cite patent |inventor-last=Fessenden |inventor-first=Reginald A. |inventorlink=Reginald Fessenden |title=Wireless Signaling |country-code=US |patent-number= 706740 |issue-date=August 12, 1902 |publication-date= September 28, 1901 }}
* {{Cite patent |inventor-last=Fessenden |inventor-first=Reginald A. |inventorlink=Reginald Fessenden |title=Method of Signaling |country-code=US |patent-number=1050728 |issue-date=January 14, 1913 |publication-date=August 21, 1906 }}

{{Authority control}}

[[Category:Frequency mixers]]
[[Category:Signal processing]]